Study About The Linear Maps Preserving Partial Isometries On B(H)

Preserving problems is always one of active research topic on operator spaces or operator algebras, and it also has important value in both theories and applications. The research of preserver problems has related to many topics, and this paper focuses on linear maps preserving partial isometries on B(H). Partial isometries are one important class of operators, and preserving partial isometries are one important class of maps on operator algebras. In recent years, many scholars have made a series of achievements in this regard. More characters of discussed maps can be found by the research. This paper contains three chapters, main content of every chapter as follows:In Chapter1, we give some notations, some definitions and some theorems which are always used in this paper. Firstly, we introduce some notations. Secondly, we give some definitions such as partial isometries, the order of partial isometries, the orthogonality of partial isometries and unitary operators and so on. Finally, some useful theorems are given.In Chapter2, we discuss the characterization of a linear map preserving partial isometries on Mn. Let Mn be the algebra of all n×n complex matrices, and let φ be a linear map on Mn(C). We proved that φ preserves partial isometries if and only if the are unitary matrices U and V such that φ(X)=UXV,(?)X∈Mn(C) or φ(X)=UXtrV,(?)X∈Mn(C), where Xtr denotes the transpose of a matrix X.In Chapter3, we mainly discuss a linear map preserving partial isometries on B(H). Let H be an infinite-dimensional complex Hilbert space and put B(H) denote the algebra of all bounded linear operators on H. Let φ is a linear bijection on B(H). It is proved that the map φ has the following structure:φ(X)=UXV or φ(X)=UXtrV, where Xtr denotes the transpose of a matrix X on some fixed base. U,V are unitary operators on H.